Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Thu, 30 Oct 2014, Penelope Maddy wrote:

I’m pretty sure Hugh would disagree with what I’m about to say, which naturally gives me pause. With that understood, I confess that from where I sit as a relatively untutored observer, it looks as if the evidence Hugh is offering is overwhelming of your Type 1 (involving the mathematical virtues of the attendant set theory).

Let me give you a counterexample.

With co-authors I established the consistency of the following

Maximality Criterion. For each infinite cardinal \alpha, \alpha^+ of \text{HOD} is less than \alpha^+.

Both Hugh and I feel that this Criterion violates the existence of certain large cardinals. If that is confirmed, then I will (tentatively) conclude that Maximality contradicts the existence of large cardinals. Hugh will conclude that there is something wrong with the above Maximality Criterion and it therefore should be rejected.

My point is that Hugh considers large cardinal existence to be part of set-theoretic truth. Why? I have yet to see an argument that large cardinal existence is needed for “good set theory”, so it does not follow from Type 1 evidence. That is why I think that large cardinal existence is part of Hugh’s personal theory of truth.

My guess is he’d also consider type 2 evidence (involving the relations of set theory to the rest of mathematics) if there were some ready to hand.

There is some ready to hand: At present, Type 2 evidence points towards Forcing Axioms, and these contradict CH and therefore contradict Ultimate L.

He has a ‘picture’ of what the set theoretic universe is like, a picture that guides his thinking, but he doesn’t expect the rest of us to share that picture and doesn’t appeal to it as a way of supporting his claims. If the mathematics goes this way rather than that, he’s quite ready to jettison a given picture and look for another. In fact, at times it seems he has several such pictures in play, interrelated by a complex system of implications (if this conjecture goes this way, the universe like this; if it goes that way, it looks like that…) But all this picturing is only heuristic, only an aide to thought — the evidence he cites is mathematical. And, yes, this is more or less how one would expect a good Thin Realist to behave (one more time: the Thin Realist also recognizes Type 2 evidence). (My apologies, Hugh. You must be thinking, with friends like these … )

That’s a lot to put in Hugh’s mouth. Probably we should invite Hugh to confirm what you say above.

The HP works quite differently. There the picture leads the way —

As with your description above, the “picture” as you call it keeps changing, even with the HP. Recall that the programme began solely with the IMH. At that time the “picture” of V was very short and fat: No inaccessibles but lots of inner models for measurable cardinals. Then came #-generation and the \textsf{IMH}^\#; a taller, handsomer universe, still with a substantial waistline. As we learn more about maximality, we refine this “picture”.

the only legitimate evidence is Type 3. As we’ve determined over the months, in this case the picture involved has to be shared, so that it won’t degenerate into ‘Sy’s truth’. So far, to be honest, I’m still not clear on the HP picture, either in its height potentialist/width actualist form or its full multiverse form. Maybe Peter is doing better than I am on that.

I have offered to work with the height potentialist/width actualist form, and even drop the reduction to ctm’s, to make people happy (this doesn’t affect the mathematical conclusions of the programme). Regarding Peter: Unless he chooses to be more open-minded, what I hear from him is a premature pessimism about the HP based on a claim that there will be “no convergence regarding what can be inferred from the maximal iterative conception”. To be honest, I find it quite odd that (excluding my coworkers Claudio and Radek) I have received the most encouragement from Hugh, who seems open-minded and interested in seeing what comes out of the HP, just as we all want to see what comes out of Ultimate L (my criticisms long ago had nothing to do with the programme itself, only with the way it had been presented).

Pen, I know that you have said that in any event you will encourage the “good set theory” that comes out of the HP. But the persistent criticism (not just from you) of the conceptual approach, aside from the math, while initially of extraordinary value to help me clarify the approach (I am grateful to you for that), is now becoming somewhat tiresome. I have written dozens of e-mails to explain what I am doing and I take it as a good sign that I am still standing, having responded consistently to each point. If there is something genuinely new to be said, fine, I will respond to it, but as I see it now we have covered everything: The HP is simply a focused investigation of mathematical criteria for the maximality of V in height and width, with the aim of convergence towards an optimal such criterion. The success of the programme will be judged by the extent to which it achieves that goal. Interesting math has already come out of the programme and will continue to come out of it. I am glad that at least Hugh has offered a bit of encouragement to me to get to work on it.


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